Chapter three Two-dimensional Cascades Laith Batarseh
Turbo cascades The linear cascade of blades comprises a number of identical blades, equally spaced and parallel to one another cascade tunnel low-speed, operating in the range 20 60 m/s high-speed, for the compressible flow range of testing
Blade nomenclature Camber line function, y Camber line Thickness, t Maximum camber, b Leading edge Trailing edge Chord line
Cascade nomenclature Stagger angle Space Blade entry angle Camber angle Blade exit angle
Useful parameter Camber line shape: mainly, circular or parabolic arc Type of thickness distribution: (t(x)) also, the following ratios Space chord ratio (s/l) t max l b l a l α 1 and α 2 Camber angle (θ =α 1 - α 2 ) Stagger angle [ξ=½(α 1 + α 2 )]
Cascade forces Two-dimensional Cascades
Cascade forces Applying the principle of continuity to a unit depth of span and noting the assumption of incompressibility, yields The momentum equation applied in the x and y directions with constant axial velocity gives,
Energy losses loss in total pressure
Energy losses A non-dimensional form pressure rise coefficient C p and a tangential force coefficient C f
Lift and drag Lift Drag
Lift and drag Lift and drag coefficients
Lift and drag Two-dimensional Cascades
Efficiency of a compressor cascade Assuming a constant lift drag ratio and differentiated with respect α m to m to give the optimum mean flow angle for maximum efficiency
Efficiency of a compressor cascade Howell (1945)
Compressor cascade performance Howell (1942) Normal design Condition Twice the min total loss (stall)
Compressor cascade performance Stall separation fluid deflection incidence stall point is arbitrarily specified as the incidence at which the total pressure loss is twice the minimum loss in total pressure stall is characterized (at positive incidence) by the flow separating from the suction side of the blade surfaces Losses profile losses (friction losses): Secondary losses (force): Y s Y p C Ds C DP 1 c 2 1 c 2 2 1 2 1
Compressor cascade performance Losses profile losses (friction losses): Secondary losses (force): Y s Y p C Ds C DP 1 c 2 1 c 2 2 1 2 1 Annulus losses : Y a C Da 1 c 2 2 1 where s, H are the blade pitch and blade length respectively
Losses Compressor cascade performance Y a C Da 1 c 2 2 1 Y s C Ds 1 c 2 2 1 Y p C DP 1 c 2 2 1 gain input input losses input Y 1 K. E Tangential velocity
Compressor cascade performance Compressor cascade correlations Many experimental investigations have confirmed that the efficient performance of compressor cascade blades is limited by the growth and separation of the blade surface boundary layers. factors which can influence the growth of the blade surface boundary layers: surface velocity distribution Reynolds number inlet Mach number free-stream turbulence Unsteadiness surface roughness. Howell use the deflection (δ = α 2 α 2 ) to correlate the designed performance. We give these parameter the superscript (*) The designed (nominal) parameters are found using Howell figures mentioned before
Compressor cascade performance Compressor cascade correlations The correlation of Howell where n=0.5 for compressor cascades and n =1 for compressor inlet guide vanes δ* is relaxation factor added to the first assumption of α 2.
Compressor cascade performance Assume α* 2 = 20 o m = 0.27 = 8.1 o Assume better value of α* 2 = 28.1 o m = 0.2862 δ* = 8.6 α* 2 = 28.6 o
Compressor cascade performance From Figure 3.16, with s/l =1 and α* 2 = 28.6 o ε = α* 1 - α* 2 = 21 o α* 1 = 49 o i = α* 1 - α 1 = - 0.4 o
Compressor cascade performance Off design conditions To find the off design conditions, Howell generate the following curve. To find the off design conditions, you have to know the design conditions (*)
Compressor cascade performance
Turbine cascade performance Ainley (1948) Reaction turbine:- the fluid accelerates through the blade row and thus experiences a pressure drop during its passage impulse turbine:- no pressure change across an its blade row Notes form graph:- the reaction blades have a much wider range of low loss performance than the impulse blades the fluid outlet angle 2 remains relatively constant over the whole range of incidence for both types
Turbine cascade performance Ainley (1948)
Turbine cascade performance Ainley (1948)
Turbine cascade performance Ainley and Mathieson (1951) Total pressure loss correlations profile loss coefficient; Y p Ainley and Mathieson (1951) represent their results as graph of verses point at which where is the incidence at stalling point which is the
Turbine cascade performance Ainley and Mathieson (1951)
Turbine cascade performance Ainley and Mathieson (1951)
Turbine cascade performance Ainley and Mathieson (1951)
Turbine cascade performance For other types of blading intermediate between nozzle blades and impulse blades Secondary loss coefficient; Y s for blades of low aspect ratio, Dunham and Came (1970):- λ is a factor Z loading coefficient
Turbine cascade performance Tip clearance coefficient; Y k k : the size and nature of the clearance gap Flow outlet angle from a turbine cascade
Zweifel (1945) Turbine cascade performance Zweifel found from a number of experiments on turbine cascades that for minimum losses the value of Ψ T was approximately 0.8. Thus, for specified inlet and outlet angles the optimum space chord ratio can be estimated